Pearson Edexcel Level 3 Advanced Subsidiary GCE in Mathematics (8MA0) Pearson Edexcel Level 3 Advanced GCE in Mathematics (9MA0)

Transcription

1 Pearson Edexcel Level 3 Advanced Subsidiary GCE in Mathematics (8MA0) Pearson Edexcel Level 3 Advanced GCE in Mathematics (9MA0) First teaching from September 2017 First certification from June 2018

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3 Contents About this booklet... 5 Pure Mathematics Proof Algebra and functions Coordinate geometry in the (x, y) plane Sequences and series Trigonometry Exponentials and logarithms Differentiation Integration Numerical methods Vectors Applied Mathematics Statistical sampling Data presentation and interpretation Probability Statistical distributions Statistical hypothesis testing Quantities and units in mechanics Kinematics Kinematics Forces and Newton s laws Moments

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5 About this booklet This booklet has been produced to support mathematics teachers delivering the new Pearson Edexcel Level 3 Advanced Subsidiary and Advanced GCE in Mathematics (8MA0 & 9MA0) (first assessment summer 2018). The booklet provides additional information on all the content from the specification accredited by Ofqual in It details the content references and provides sample questions for each content topic taken from the Sample Assessment Materials. 5

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7 1. Proof Pure Mathematics 1. Proof Structure and calculation 1.1 Understand and use the structure of mathematical proof, proceeding from given assumptions through a series of logical steps to a conclusion; use methods of proof, including: Proof by deduction Proof by exhaustion Disproof by counterexample. Proof by contradiction (including proof of the irrationality of 2 and the infinity of primes, and application to unfamiliar proofs). Sample questions AS Mathematics SAMs Paper 1 Q10 (1.1, 2.3, 2.5) The equation kx 2 + 4kx + 3 = 0, where k is a constant, has no real roots. Prove that 0 k 4 3. (Total for Question 10 is 4 marks) AS Mathematics SAMs Paper 1 Q11 (1.1) (a) Prove that for all positive values of x and y, x y xy. 2 (b) Prove by counterexample that this is not true when x and y are both negative. (1) (Total for Question 11 is 3 marks) 7

8 1. Proof AS Mathematics SAMs Paper 1 Q16 (1.1, 7.1, 7.2, 7.3) A B 2x m E y m D C Figure 4 Figure 4 shows the plan view of the design for a swimming pool. The shape of this pool ABCDEA consists of a rectangular section ABDE joined to a semi-circular section BCD as shown in Figure 4. Given that AE = 2x metres, ED = y metres and the area of the pool is 250 m 2, (a) show that the perimeter, P metres, of the pool is given by P = 2x x 2x (4) 500 (b) Explain why 0 < x. (c) Find the minimum perimeter of the pool, giving your answer to 3 significant figures. (4) (Total for Question 16 is 10 marks) A level Mathematics SAMs Paper 1 Q1 (1.1, 7.1, 7.2, 7.3) The curve C has equation y = 3x 4 8x 3 3. (a) Find (i) d y, dx 2 d y (ii) 2. dx (b) Verify that C has a stationary point when x 2 (a) Determine the nature of this stationary point, giving a reason for your answer. (Total for Question 1 is 7 marks) 8

9 1. Proof A level Mathematics SAMs Paper 1 Q9 (1.1, 5.4, 5.5, 5.6, 5.8) (a) Prove that tan + cot 2 cosec 2, n, n Z. 2 (4) (b) Hence explain why the equation tan + cot 1 does not have any real solutions. (1) (Total for Question 9 is 5 marks) A level Mathematics SAMs Paper 1 Q10 (1.1, 5.6, 7.1) Given that is measured in radians, prove, from first principles, that the derivative of sin is cos You may assume the formula for sin (A B) and that, as h 0, sin h h 1 and cos h 1 0. h (Total for Question 10 is 5 marks) A level Mathematics SAMs Paper 2 Q4 (1.1, 2.8, 6.3) Given f(x) = e x, x R, g(x) = 3 ln x, x > 0, x R, (a) find an expression for gf (x), simplifying your answer. (b) Show that there is only one real value of x for which gf(x) = fg(x). (Total for Question 4 is 5 marks) 9

10 1. Proof A level Mathematics SAMs Paper 2 Q6 (1.1, 2.3, 2.5) For each statement below you must state if it is always true, sometimes true or never true, giving a reason in each case. The first one has been done for you. Statement Always True Sometimes True Never True Reason The quadratic equation ax 2 + bx + c = 0 (a 0) has 2 real roots. (i) When a real value of x is substituted into x 2 6x + 10 the result is positive. It only has 2 real roots when b 2 4ac > 0 When b 2 4ac = 0 it has 1 real root and when b 2 4ac < 0 it has 0 real roots. (ii) If ax > b then x a b (iii) The difference between consecutive square numbers is odd. (Total for Question 6 is 6 marks) A level Mathematics SAMs Paper 2 Q14 (1.1, 2.11, 7.2, 7.3) A company decides to manufacture a soft drinks can with a capacity of 500 ml. The company models the can in the shape of a right circular cylinder with radius r cm and height h cm. In the model they assume that the can is made from a metal of negligible thickness. (a) Prove that the total surface area, S cm 2, of the can is given by S = 2 r 2 + Given that r can vary, r (b) find the dimensions of a can that has minimum surface area. (5) (c) With reference to the shape of the can, suggest a reason why the company may choose not to manufacture a can with minimum surface area. (1) (Total for Question 14 is 9 marks) 10

11 2. Algebra and functions 2. Algebra and functions Structure and calculation 2.1 Understand and use the laws of indices for all rational exponents. 2.2 Use and manipulate surds, including rationalising the denominator. 2.3 Work with quadratic functions and their graphs. The discriminant of a quadratic function, including the conditions for real and repeated roots. Completing the square. Solution of quadratic equations including solving quadratic equations in a function of the unknown. 2.4 Solve simultaneous equations in two variables by elimination and by substitution, including one linear and one quadratic equation. 2.5 Solve linear and quadratic inequalities in a single variable and interpret such inequalities graphically, including inequalities with brackets and fractions. Express solutions through correct use of and and or, or through set notation. Represent linear and quadratic inequalities such as y > x + 1 and y > ax 2 + bx + c graphically. 2.6 Manipulate polynomials algebraically, including expanding brackets and collecting like terms, factorisation and simple algebraic division; use of the factor theorem. Simplify rational expressions, including by factorising and cancelling, and algebraic division (by linear expressions only). 2.7 Understand and use graphs of functions; sketch curves defined by simple equations including polynomials. The modulus of a linear function. a a Y = and y = x 2 x (including their vertical and horizontal asymptotes) Interpret algebraic solution of equations graphically; use intersection points of graphs to solve equations. Understand and use proportional relationships and their graphs. 2.8 Understand and use composite functions; inverse functions and their graphs. 2.9 Understand the effect of simple transformations on the graph of y = f(x), including sketching associated graphs: y = af(x), y = f(x) + a, y = f(x + a), y = f(ax) and combinations of these transformations Decompose rational functions into partial fractions (denominators not more complicated than squared linear terms and with no more than 3 terms, numerators constant or linear) Use of functions in modelling, including consideration of limitations and refinements of the models. 11

12 2. Algebra and functions Sample questions AS Mathematics SAMs Paper 1 Q3 (2.1, 2.2, 8.2, 8.3) Given that the point A has position vector 3i 7j and the point B has position vector 8i + 3j, (a) find the vector AB. (b) Find AB. Give your answer as a simplified surd. (Total for Question 3 is 4 marks) AS Mathematics SAMs Paper 1 Q4 (2.3, 2.6) f(x) = 4x 3 12x 2 + 2x 6 (a) Use the factor theorem to show that (x 3) is a factor of f(x). (b) Hence show that 3 is the only real root of the equation f(x) = 0. (4) (Total for Question 4 is 6 marks) AS Mathematics SAMs Paper 1 Q5 (1.1, 2.3, 2.5) 12 Given that f(x) = 2x x, x > 0, 2 2 show that 1 f ( x ) dx = (Total for Question 5 is 5 marks) AS Mathematics SAMs Paper 1 Q10 (1.1, 2.3, 2.5) The equation kx 2 + 4kx + 3 = 0, where k is a constant, has no real roots. Prove that 0 k 4 3. (Total for Question 10 is 4 marks) 12

13 2. Algebra and functions AS Mathematics SAMs Paper 1 Q12 (2.1, 6.5) A student was asked to give the exact solution to the equation 2 2x + 4 9(2 x ) = 0. The student s attempt is shown below: 2 2x + 4 9(2 x ) = 0 2 2x (2 x ) = 0 Let 2 x = y y 2 9y + 8 = 0 (y 8)(y 1) = 0 y = 8 or y = 1 So x = 3 or x 0 (a) Identify the two errors made by the student. (b) Find the exact solution to the equation. (Total for Question 12 is 4 marks) AS Mathematics SAMs Paper 1 Q13 (2.1, 6.5) (a) Factorise completely x x x. (b) Sketch the curve with equation y = x x x, showing the coordinates of the points at which the curve cuts or touches the x-axis. The point with coordinates ( 3, 0) lies on the curve with equation y = (x + a) (x + a) (x + a), where a is a constant. (c) Find the two possible values of a. (Total for Question 13 is 7 marks) 13

14 2. Algebra and functions AS Mathematics SAMs Paper 1 Q15 (2.4, 6.3, 7.2, 7.3) y C1 Diagram not drawn to scale C2 O x The curve C1, shown in Figure 3, has equation y = 4x 2 6x The point P, 2 2 lies on C1. The curve C2, also shown in Figure 3, has equation y = 21 x ln (2x). The normal to C1 at the point P meets C2 at the point Q. Find the exact coordinates of Q. (Solutions based entirely on graphical or numerical methods are not acceptable.) (Total for Question 15 is 8 marks) 14

15 2. Algebra and functions A level Mathematics SAMs Paper 1 Q3 (2.3, 2.5, 3.2) A circle C has equation x 2 + y 2 4x + 10y = k, where k is a constant. (a) Find the coordinates of the centre of C. (b) State the range of possible values for k. (Total for Question 3 is 4 marks) A level Mathematics SAMs Paper 1 Q5 (2.6, 3.3) A curve C has parametric equations x = 2t 1, y = 4t 7 + t 3, t 0. Show that the Cartesian equation of the curve C can be written in the form y = 2x 2 ax b, x 1, x 1 where a and b are integers to be found. (Total for Question 5 is 3 marks) 15

16 2. Algebra and functions A level Mathematics SAMs Paper 1 Q6 (2.11, 3.1, 6.1, 6.3) A car dealer wishes to model the value of a certain type of car. The initial price of the car is and the value after 4 years is expected to be In a simple model, the value of the car, V, depends only on the age of the car, t years. The diagram below shows the graphs of two possible models over 12 years. (a) Explain why Model A is an unrealistic model for cars over 10 years of age. It is given that the equation for Model B is of the form V = pe kt, where p and k are constants. (b) Find the equation for Model B. Saima wants to know the value of her car when it is 3 years old. (c) (i) Use Model A to predict the value of Saima s car. (ii) Use Model B to predict the value of Saima s car. (d) Write down one possible refinement of either Model A or Model B. (5) (Total for Question 6 is 7 marks) 16

17 2. Algebra and functions A level Mathematics SAMs Paper 1 Q8 (2.7, 9.1, 9.2) f(x) = ln (2x 5) + 2x 2 30, x > 2.5. (a) Show that f(x) 0 has a root in the interval 3.5, 4. A student takes 4 as the first approximation to. Given f(4) = and f (4) = to 4 significant figures, (b) apply the Newton-Raphson procedure once to obtain a second approximation for, giving your answer to 3 significant figures. (c) Show that is the only root of f(x) 0. (Total for Question 8 is 6 marks) A level Mathematics SAMs Paper 1 Q11 (2.3, 2.11) An archer shoots an arrow. The height, H metres, of the arrow above the ground is modelled by the formula H d 0.002d 2, d 0, where d is the horizontal distance of the arrow from the archer, measured in metres. Given that the arrow travels in a vertical plane until it hits the ground, (a) find the horizontal distance travelled by the arrow, as given by this model. (b) With reference to the model, interpret the significance of the constant 1.8 in the formula. (c) Write d 0.002d 2 in the form A B(d C) 2, where A, B and C are constants to be found. It is decided that the model should be adapted for a different archer. The adapted formula for this archer is H d 0.002d 2, d 0. Hence, or otherwise, find, for the adapted model, (d) (i) the maximum height of the arrow above the ground. (ii) the horizontal distance, from the archer, of the arrow when it is at its maximum height. (Total for Question 11 is 9 marks) (1) 17

18 2. Algebra and functions A level Mathematics SAMs Paper 1 Q13 (2.2, 3.1, 3.3, 5.3, 5.6, 7.2, 7.3, 7.5) The curve C has parametric equations x = 2 cos t, y = 3 cos 2t, 0 t π. (a) Find an expression for The point P lies on C where t dy in terms of t. dx 2. 3 The line l is the normal to C at P. (b) Show that an equation for l is 2x (2 3)y 1 = 0. The line l intersects the curve C again at the point Q. (5) (c) Find the exact coordinates of Q. You must show clearly how you obtained your answers. (6) (Total for Question 13 is 13 marks) 18

19 2. Algebra and functions A level Mathematics SAMs Paper 1 Q15 (2.9, 5.4, 7.2, 7.3, 7.4) Figure 5 Figure 5 shows a sketch of the curve with equation y f(x), where f(x) = 4sin 2x, 0 x e 2x 1 The curve has a maximum turning point at P and a minimum turning point at Q, as shown in Figure 5. (a) Show that the x-coordinates of point P and point Q are solutions of the equation tan 2x = 2. (b) Using your answer to part (a), find the x-coordinate of the minimum turning point on the curve with equation (i) y f(2x), (ii) y 3 2f(x). (Total for Question 15 is 8 marks) A level Mathematics SAMs Paper 2 Q1 (2.6) f(x) = 2x 3 5x 2 + ax + a. Given that x + 2 is a factor of f (x), find the value of the constant a. (Total for Question 1 is 3 marks) 19

20 2. Algebra and functions A level Mathematics SAMs Paper 2 Q4 (1.1, 2.8, 6.3) Given f(x) = e x, x R, g(x) = 3 ln x, x > 0, x R, (a) find an expression for gf (x), simplifying your answer. (b) Show that there is only one real value of x for which gf(x) = fg(x). (Total for Question 4 is 5 marks) A level Mathematics SAMs Paper 2 Q6 (1.1, 2.3, 2.5) For each statement below you must state if it is always true, sometimes true or never true, giving a reason in each case. The first one has been done for you. Statement Always True Sometimes True Never True Reason The quadratic equation ax 2 + bx + c = 0 (a 0) has 2 real roots. (i) When a real value of x is substituted into x 2 6x + 10 the result is positive. It only has 2 real roots when b 2 4ac > 0 When b 2 4ac = 0 it has 1 real root and when b 2 4ac < 0 it has 0 real roots. (ii) If ax > b then x a b (iii) The difference between consecutive square numbers is odd. (Total for Question 6 is 6 marks) 20

21 2. Algebra and functions A level Mathematics SAMs Paper 2 Q9 (2.3, 8.2, 8.3) Given that A is constant and 1 4 ( 3 x A) dx = 2A 2, show that there are exactly two possible values for A. (Total for question 9 is 5 marks) A level Mathematics SAMs Paper 2 Q11 (2.5, 2.7, 2.9) Figure 2 Figure 2 shows a sketch of part of the graph y = f(x) where f(x) = 2 3 x + 5, x 0. (a) State the range of f. (1) 1 (b) Solve the equation f(x) x Given that the equation f(x) = k, where k is a constant, has two distinct roots, (c) state the set of possible values for k. (Total for Question 11 is 6 marks) 21

22 2. Algebra and functions A level Mathematics SAMs Paper 2 Q12 (2.3, 5.3, 5.5, 5.7) (a) Solve, for 180 x < 180, the equation 3 sin 2 x + sin x cos 2 x, giving your answers to 2 decimal places. (6) (b) Hence find the smallest positive solution of the equation 3 sin 2 (2 30 ) + sin (2 30 ) + 8 = 9 cos 2 (2 30 ), giving your answer to 2 decimal places. (Total for Question 12 is 8 marks) A level Mathematics SAMs Paper 2 Q13 (5.3, 5.6, 5.9) (a) Express 10 cos 3 sin in the form R cos ( + ), where R > 0 and 0 < < 90. Give the exact value of R and give the value of, in degrees, to 2 decimal places. Figure 3 The height above the ground, H metres, of a passenger on a Ferris wheel t minutes after the wheel starts turning, is modelled by the equation H = a 10 cos (80t) + 3 sin (80t), where a is a constant. Figure 3 shows the graph of H against t for two complete cycles of the wheel. Given that the initial height of the passenger above the ground is 1 metre, (b) (i) find a complete equation for the model. (ii) Hence find the maximum height of the passenger above the ground. (c) Find the time taken, to the nearest second, for the passenger to reach the maximum height on the second cycle. (Solutions based entirely on graphical or numerical methods are not acceptable.) It is decided that, to increase profits, the speed of the wheel is to be increased. (d) How would you adapt the equation of the model to reflect this increase in speed? (1) (Total for Question 13 is 9 marks) 22

23 2. Algebra and functions A level Mathematics SAMs Paper 2 Q14 (1.1, 2.11, 7.2, 7.3) A company decides to manufacture a soft drinks can with a capacity of 500 ml. The company models the can in the shape of a right circular cylinder with radius r cm and height h cm. In the model they assume that the can is made from a metal of negligible thickness. (a) Prove that the total surface area, S cm 2, of the can is given by S = 2 r 2 + Given that r can vary, r (b) find the dimensions of a can that has minimum surface area. (5) (c) With reference to the shape of the can, suggest a reason why the company may choose not to manufacture a can with minimum surface area. (1) (Total for Question 14 is 9 marks) A level Mathematics SAMs Paper 2 Q16 (2.1, 6.3, 6.4, 8.6, 8.7, 8.8) 1 (a) Express in partial fractions. P(11 2P) A population of meerkats is being studied. The population is modelled by the differential equation dp 1 = P(11 2P), t 0, 0 < P < 5.5, dt 22 where P, in thousands, is the population of meerkats and t is the time measured in years since the study began. Given that there were 1000 meerkats in the population when the study began, (b) determine the time taken, in years, for this population of meerkats to double, (c) show that A P = B C t 2 e 1, (6) where A, B and C are integers to be found. (Total for Question 16 is 12 marks) 23

24 3. Coordinate geometry in the (x, y) plane 3. Coordinate geometry in the (x, y) plane Structure and calculation 3.1 Understand and use the equation of a straight line, including the forms y y 1 = m(x x 1) and ax + by + c = 0; Gradient conditions for two straight lines to be parallel or perpendicular. Be able to use straight line models in a variety of contexts. 3.2 Understand and use the coordinate geometry of the circle including using the equation of a circle in the form (x a) 2 + (y b) 2 = r 2 Completing the square to find the centre and radius of a circle; use of the following properties: the angle in a semicircle is a right angle the perpendicular from the centre to a chord bisects the chord the radius of a circle at a given point on its circumference is perpendicular to the tangent to the circle at that point. 3.3 Understand and use the parametric equations of curves and conversion between Cartesian and parametric forms. 3.4 Use parametric equations in modelling in a variety of contexts. Sample questions AS Mathematics SAMs Paper 1 Q1 (3.1) The line l passes through the points A (3, 1) and B (4, 2). Find an equation for l. (Total for Question 1 is 3 marks) 24

25 3. Coordinate geometry in the (x, y) plane AS Mathematics SAMs Paper 1 Q14 (3.1, 6.4, 6.6, 6.7) log10 P l (0, 5) O Figure 2 t A town s population, P, is modelled by the equation P = ab t, where a and b are constants and t is the number of years since the population was first recorded. The line l shown in Figure 2 illustrates the linear relationship between t and log10 P for the population over a period of 100 years. 1 The line l meets the vertical axis at (0, 5) as shown. The gradient of l is. 200 (a) Write down an equation for l. (b) Find the value of a and the value of b. (c) With reference to the model, interpret (4) (i) the value of the constant a, (ii) the value of the constant b. (d) Find (i) the population predicted by the model when t = 100, giving your answer to the nearest hundred thousand, (ii) the number of years it takes the population to reach , according to the model. (e) State two reasons why this may not be a realistic population model. (Total for Question 14 is 13 marks) 25

26 3. Coordinate geometry in the (x, y) plane AS Mathematics SAMs Paper 1 Q17 (3.2) A circle C with centre at ( 2, 6) passes through the point (10, 11). (a) Show that the circle C also passes through the point (10, 1). The tangent to the circle C at the point (10, 11) meets the y-axis at the point P and the tangent to the circle C at the point (10, 1) meets the y-axis at the point Q. (b) Show that the distance PQ is 58, explaining your method clearly. (7) (Total for Question 17 is 10 marks) A level Mathematics SAMs Paper 1 Q3 (2.3, 2.5, 3.2) A circle C has equation x 2 + y 2 4x + 10y = k, where k is a constant. (a) Find the coordinates of the centre of C. (b) State the range of possible values for k. (Total for Question 3 is 4 marks) A level Mathematics SAMs Paper 1 Q5 (2.6, 3.3) A curve C has parametric equations x = 2t 1, y = 4t 7 + t 3, t 0. Show that the Cartesian equation of the curve C can be written in the form y = 2x 2 ax b, x 1, x 1 where a and b are integers to be found. (Total for Question 5 is 3 marks) 26

27 3. Coordinate geometry in the (x, y) plane A level Mathematics SAMs Paper 1 Q6 (2.11, 3.1, 6.1, 6.3) A car dealer wishes to model the value of a certain type of car. The initial price of the car is and the value after 4 years is expected to be In a simple model, the value of the car, V, depends only on the age of the car, t years. The diagram below shows the graphs of two possible models over 12 years. (a) Explain why Model A is an unrealistic model for cars over 10 years of age. It is given that the equation for Model B is of the form V = pe kt, where p and k are constants. (b) Find the equation for Model B. Saima wants to know the value of her car when it is 3 years old. (c) (i) Use Model A to predict the value of Saima s car. (ii) Use Model B to predict the value of Saima s car. (d) Write down one possible refinement of either Model A or Model B. (5) (Total for Question 6 is 7 marks) 27

28 3. Coordinate geometry in the (x, y) plane A level Mathematics SAMs Paper 1 Q13 (2.2, 3.1, 3.3, 5.3, 5.6, 7.2, 7.3, 7.5) The curve C has parametric equations x = 2 cos t, y = 3 cos 2t, 0 t π. (a) Find an expression for The point P lies on C where t dy in terms of t. dx 2. 3 The line l is the normal to C at P. (b) Show that an equation for l is 2x (2 3)y 1 = 0. The line l intersects the curve C again at the point Q. (5) (c) Find the exact coordinates of Q. You must show clearly how you obtained your answers. (6) (Total for Question 13 is 13 marks) A level Mathematics SAMs Paper 2 Q8 (3.1) Figure 1 Figure 1 shows a rectangle ABCD. The point A lies on the y-axis and the points B and D lie on the x-axis as shown in Figure 1. Given that the straight line through the points A and B has equation 5y + 2x = 10, (a) show that the straight line through the points A and D has equation 2y 5x = 4, (4) (b) find the area of the rectangle ABCD. (Total for Question 8 is 7 marks) 28

29 4. Sequences and series 4. Sequences and series Structure and calculation 4.1 Understand and use the binomial expansion of (a + bx) n for positive integer n; the notations n! and n C r link to binomial probabilities. Extend to any rational n, including its use for approximation; be aware that the expansion is valid bx for < 1 (proof not required) a 4.2 Work with sequences including those given by a formula for the nth term and those generated by a simple relation of the form x n + 1 = f(x n); increasing sequences; decreasing sequences; periodic sequences. 4.3 Understand and use sigma notation for sums of series. 4.4 Understand and work with arithmetic sequences and series, including the formulae for nth term and the sum to n terms. 4.4 Understand and work with arithmetic sequences and series, including the formulae for nth term and the sum to n terms. 4.5 Understand and work with geometric sequences and series, including the formulae for the nth term and the sum of a finite geometric series; the sum to infinity of a convergent geometric series, including the use of r < 1; modulus notation 4.6 Use sequences and series in modelling. Sample questions AS Mathematics SAMs Paper 1 Q7 (4.1) (a) Find the first 3 terms, in ascending powers of x, of the binomial expansion of 2 x, giving 2 each term in its simplest form. (4) (b) Explain how you would use your expansion to give an estimate for the value of (1) (Total for Question 7 is 5 marks) 7 29

30 4. Sequences and series A level Mathematics SAMs Paper 2 Q7 (4.1) (a) Use the binomial expansion, in ascending powers of x, to show that (4 x) = x kx + where k is a rational constant to be found. (4) A student attempts to substitute x 1 into both sides of this equation to find an approximate value for 3. (b) State, giving a reason, if the expansion is valid for this value of x. (1) (Total for Question 7 is 5 marks) A level Mathematics SAMs Paper 2 Q10 (4.5) In a geometric series the common ratio is r and sum to n terms is Sn. Given S = 7 8 S6, show that 1 1, where k is an integer to be found. k (Total for Question 10 is 4 marks) 30

31 5. Trigonometry 5. Trigonometry Structure and calculation 5.1 Understand and use the definitions of sine, cosine and tangent for all arguments; the sine and cosine rules; the area of a triangle in the form 2 1 ab sin C Work with radian measure, including use for arc length and area of sector. 5.2 Understand and use the standard small angle approximations of sine, cosine and tangent sin θ θ, 2 cos θ = 1, tan θ θ Where θ is in radians Understand and use the sine, cosine and tangent functions; their graphs, symmetries and periodicity. Know and use exact values of sin and cos for 0, 6, 4, 3, 2, π and multiples thereof, and exact values of tan for 0, 6, 4, 3, 2, π and multiples thereof. 5.4 Understand and use the definitions of secant, cosecant and cotangent and of arcsin, arccos and arctan; their relationships to sine, cosine and tangent; understanding of their graphs; their ranges and domains. 5.5 Understand and use tan θ = sin cos Understand and use sin 2 θ + cos 2 θ = 1, sec 2 θ = 1 + tan 2 θ and cosec 2 θ = 1+ cot 2 θ 5.6 Understand and use double angle formulae; use of formulae for sin (A ± B), cos (A ± B) and tan (A ± B), understand geometrical proofs of these formulae. Understand and use expressions for a cos θ + b sin θ in the equivalent forms of r cos (θ ± α) or r sin (θ ± α) 5.7 Solve simple trigonometric equations in a given interval, including quadratic equations in sin, cos and tan and equations involving multiples of the unknown angle. 5.8 Construct proofs involving trigonometric functions and identities. 5.9 Use trigonometric functions to solve problems in context, including problems involving vectors, kinematics and forces. 31

32 5. Trigonometry Sample questions AS Mathematics SAMs Paper 1 Q8 (5.1) C A m 60 B Figure 1 A triangular lawn is modelled by the triangle ABC, shown in Figure 1. The length AB is to be 30 m long. Given that angle BAC = 70 and angle ABC = 60, (a) calculate the area of the lawn to 3 significant figures. (b) Why is your answer unlikely to be accurate to the nearest square metre? (Total for Question 8 is 5 marks) AS Mathematics SAMs Paper 1 Q9 (5.3, 5.4) Solve, for 360 x < 540, Give your answers to one decimal place. 12 sin 2 x + 7 cos x 13 = 0. (Solutions based entirely on graphical or numerical methods are not acceptable.) (Total for Question 9 is 5 marks) 32

33 5. Trigonometry A level Mathematics SAMs Paper 1 Q2 (5.1) Figure 1 The shape ABCDOA, as shown in Figure 1, consists of a sector COD of a circle centre O joined to a sector AOB of a different circle, also centre O. Given that arc length CD = 3 cm, COD = 0.4 radians and AOD is a straight line of length 12 cm, find (a) the length of OD, (b) the area of the shaded sector AOB. (Total for Question 2 is 5 marks) A level Mathematics SAMs Paper 1 Q7 (5.1, 10.1, 10.2, 10.3) Figure 2 shows a sketch of a triangle ABC. Figure 2 Given AB = 2i + 3j + k and BC = i 9j + 3k, show that BAC = to one decimal place. (Total for Question 7 is 5 marks) 33

34 5. Trigonometry A level Mathematics SAMs Paper 1 Q9 (1.1, 5.4, 5.5, 5.6, 5.8) (a) Prove that tan + cot 2 cosec 2, n, n Z. 2 (4) (b) Hence explain why the equation tan + cot 1 does not have any real solutions. (1) (Total for Question 9 is 5 marks) A level Mathematics SAMs Paper 1 Q10 (1.1, 5.6, 7.1) Given that is measured in radians, prove, from first principles, that the derivative of sin is cos You may assume the formula for sin (A B) and that, as h 0, sin h h 1 and cos h 1 0. h (Total for Question 10 is 5 marks) A level Mathematics SAMs Paper 1 Q13 (2.2, 3.1, 3.3, 5.3, 5.6, 7.2, 7.3, 7.5) The curve C has parametric equations x = 2 cos t, y = 3 cos 2t, 0 t π. (a) Find an expression for The point P lies on C where t dy in terms of t. dx 2. 3 The line l is the normal to C at P. (b) Show that an equation for l is 2x (2 3)y 1 = 0. The line l intersects the curve C again at the point Q. (5) (c) Find the exact coordinates of Q. You must show clearly how you obtained your answers. (6) (Total for Question 13 is 13 marks) 34

35 5. Trigonometry A level Mathematics SAMs Paper 1 Q15 (2.9, 5.4, 7.2, 7.3, 7.4) Figure 5 Figure 5 shows a sketch of the curve with equation y f(x), where f(x) = 4sin 2x, 0 x e 2x 1 The curve has a maximum turning point at P and a minimum turning point at Q, as shown in Figure 5. (a) Show that the x-coordinates of point P and point Q are solutions of the equation tan 2x = 2. (4) (b) Using your answer to part (a), find the x-coordinate of the minimum turning point on the curve with equation (i) y f(2x), (ii) y 3 2f(x). (4) (Total for Question 15 is 8 marks) 35

36 5. Trigonometry A level Mathematics SAMs Paper 2 Q2 (5.5, 5.7) Some A level students were given the following question. Solve, for 90 90, the equation cos = 2 sin. The attempts of two of the students are shown below. Student A cos = 2 sin tan = 2 = 63.4 Student B cos = 2 sin cos 2 = 4 sin 2 1 sin 2 = 4 sin 2 sin 2 = sin = 5 = 26.6 (a) Identify an error made by student A. (1) Student B gives 26.6 as one of the answers to cos = 2 sin. (b) (i) Explain why this answer is incorrect. (ii) Explain how this incorrect answer arose. (Total for Question 2 is 3 marks) A level Mathematics SAMs Paper 2 Q12 (2.3, 5.3, 5.5, 5.7) (a) Solve, for 180 x < 180, the equation 3 sin 2 x + sin x cos 2 x, giving your answers to 2 decimal places. (6) (b) Hence find the smallest positive solution of the equation 3 sin 2 (2 30 ) + sin (2 30 ) + 8 = 9 cos 2 (2 30 ), giving your answer to 2 decimal places. (Total for Question 12 is 8 marks) 36

37 5. Trigonometry A level Mathematics SAMs Paper 2 Q13 (5.3, 5.6, 5.9) (a) Express 10 cos 3 sin in the form R cos ( + ), where R > 0 and 0 < < 90. Give the exact value of R and give the value of, in degrees, to 2 decimal places. Figure 3 The height above the ground, H metres, of a passenger on a Ferris wheel t minutes after the wheel starts turning, is modelled by the equation H = a 10 cos (80t) + 3 sin (80t), where a is a constant. Figure 3 shows the graph of H against t for two complete cycles of the wheel. Given that the initial height of the passenger above the ground is 1 metre, (b) (i) find a complete equation for the model. (ii) Hence find the maximum height of the passenger above the ground. (c) Find the time taken, to the nearest second, for the passenger to reach the maximum height on the second cycle. (Solutions based entirely on graphical or numerical methods are not acceptable.) It is decided that, to increase profits, the speed of the wheel is to be increased. (d) How would you adapt the equation of the model to reflect this increase in speed? (1) (Total for Question 13 is 9 marks) 37

38 6. Exponentials and logarithms 6. Exponentials and logarithms Structure and calculation 6.1 Know and use the function a x and its graph, where a is positive. Know and use the function e x and its graph. 6.2 Know that the gradient of e kx is equal to ke kx and hence understand why the exponential model is suitable in many applications. 6.3 Know and use the definition of log a x as the inverse of a x, where a is positive and x 0. Know and use the function ln x and its graph. Know and use ln x as the inverse function of e x 6.4 Understand and use the laws of logarithms: log a x + log a y = log a (xy) log a x log a y = log a y x k log a x = log a x k (including, for example, k = 1 and k = 2 1 ) 6.5 Solve equations of the form a x = b 6.6 Use logarithmic graphs to estimate parameters in relationships of the form y = ax n and y = kb x, given data for x and y 6.7 Understand and use exponential growth and decay; use in modelling (examples may include the use of e in continuous compound interest, radioactive decay, drug concentration decay, exponential growth as a model for population growth); consideration of limitations and refinements of exponential models. 38

39 6. Exponentials and logarithms Sample questions AS Mathematics SAMs Paper 1 Q12 (2.1, 6.5) A student was asked to give the exact solution to the equation 2 2x + 4 9(2 x ) = 0. The student s attempt is shown below: 2 2x + 4 9(2 x ) = 0 2 2x (2 x ) = 0 Let 2 x = y y 2 9y + 8 = 0 (y 8)(y 1) = 0 y = 8 or y = 1 So x = 3 or x 0 (a) Identify the two errors made by the student. (b) Find the exact solution to the equation. (Total for Question 12 is 4 marks) AS Mathematics SAMs Paper 1 Q13 (2.1, 6.5) (a) Factorise completely x x x. (b) Sketch the curve with equation y = x x x, showing the coordinates of the points at which the curve cuts or touches the x-axis. The point with coordinates ( 3, 0) lies on the curve with equation y = (x + a) (x + a) (x + a), where a is a constant. (c) Find the two possible values of a. (Total for Question 13 is 7 marks) 39

40 6. Exponentials and logarithms AS Mathematics SAMs Paper 1 Q14 (3.1, 6.4, 6.6, 6.7) log10 P l (0, 5) O Figure 2 t A town s population, P, is modelled by the equation P = ab t, where a and b are constants and t is the number of years since the population was first recorded. The line l shown in Figure 2 illustrates the linear relationship between t and log10 P for the population over a period of 100 years. 1 The line l meets the vertical axis at (0, 5) as shown. The gradient of l is. 200 (a) Write down an equation for l. (b) Find the value of a and the value of b. (c) With reference to the model, interpret (4) (i) the value of the constant a, (ii) the value of the constant b. (d) Find (i) the population predicted by the model when t = 100, giving your answer to the nearest hundred thousand, (ii) the number of years it takes the population to reach , according to the model. (e) State two reasons why this may not be a realistic population model. (Total for Question 14 is 13 marks) 40

41 6. Exponentials and logarithms AS Mathematics SAMs Paper 1 Q15 (2.4, 6.3, 7.2, 7.3) y C1 Diagram not drawn to scale C2 O x The curve C1, shown in Figure 3, has equation y = 4x 2 6x The point P, 2 2 lies on C1. The curve C2, also shown in Figure 3, has equation y = 21 x ln (2x). The normal to C1 at the point P meets C2 at the point Q. Find the exact coordinates of Q. (Solutions based entirely on graphical or numerical methods are not acceptable.) (Total for Question 15 is 8 marks) A level Mathematics SAMs Paper 1 Q4 (6.4, 8.2, 8.3) Given that a is a positive constant and 2a a t 1 dt t = ln 7, show that a ln k, where k is a constant to be found. (Total for Question 4 is 4 marks) 41

42 6. Exponentials and logarithms A level Mathematics SAMs Paper 1 Q6 (2.11, , 6.3) A car dealer wishes to model the value of a certain type of car. The initial price of the car is and the value after 4 years is expected to be In a simple model, the value of the car, V, depends only on the age of the car, t years. The diagram below shows the graphs of two possible models over 12 years. (a) Explain why Model A is an unrealistic model for cars over 10 years of age. It is given that the equation for Model B is of the form V = pe kt, where p and k are constants. (b) Find the equation for Model B. Saima wants to know the value of her car when it is 3 years old. (c) (i) Use Model A to predict the value of Saima s car. (ii) Use Model B to predict the value of Saima s car. (d) Write down one possible refinement of either Model A or Model B. (5) (Total for Question 6 is 7 marks) 42

43 6. Exponentials and logarithms A level Mathematics SAMs Paper 1 Q12 (6.1, 6.3, 6.4, 6.6, 6.7) In a controlled experiment, the number of microbes, N, present in a culture T days after the start of the experiment, were counted. N and T are expected to satisfy a relationship of the form N = at b, where a and b are constants. (a) Show that this relationship can be expressed in the form log10 N = m log10 T + c, giving m and c in terms of the constants a and/or b. Figure 3 shows the line of best fit for values of log10 N plotted against values of log10 T. (b) Use the information provided to estimate the number of microbes present in the culture 3 days after the start of the experiment. (4) (c) Explain why the information provided could not reliably be used to estimate the day when the number of microbes in the culture first exceeds (d) With reference to the model, interpret the value of the constant a. (1) (Total for Question 12 is 9 marks) 43

44 6. Exponentials and logarithms A level Mathematics SAMs Paper 1 Q14 (6.4, 8.3, 8.5, 9.3) Figure 4 Figure 4 shows a sketch of part of the curve C with equation y = x 2 ln 3 x 2x + 5, x > 0 The finite region S, shown shaded in Figure 4, is bounded by the curve C, the line with equation x = 1, the x-axis and the line with equation x = 3. The table below shows corresponding values of x and y, with the values of y given to 4 decimal places as appropriate. x y (a) Use the trapezium rule, with all the values of y in the table, to obtain an estimate for the area of S, giving your answer to 3 decimal places. (b) Explain how the trapezium rule could be used to obtain a more accurate estimate for the area of S. (1) (c) Show that the exact area of S can be written in the form b a + ln c, where a, b and c are integers to be found. (6) (In part (c), solutions based entirely on graphical or numerical methods are not acceptable.) (Total for Question 14 is 10 marks) 44

45 6. Exponentials and logarithms A level Mathematics SAMs Paper 2 Q4 (1.1, 2.8, 6.3) Given f(x) = e x, x R, g(x) = 3 ln x, x > 0, x R, (a) find an expression for gf (x), simplifying your answer. (b) Show that there is only one real value of x for which gf(x) = fg(x). (Total for Question 4 is 5 marks) A level Mathematics SAMs Paper 2 Q5 (6.2, 6.7) The mass, m grams, of a radioactive substance, t years after first being observed, is modelled by the equation m = 25e 0.05t. According to the model, (a) find the mass of the radioactive substance six months after it was first observed, dm (b) show that = km, where k is a constant to be found. dt (Total for Question 5 is 4 marks) 45

46 6. Exponentials and logarithms A level Mathematics SAMs Paper 2 Q16 (2.1, 6.3, 6.4, 8.6, 8.7, 8.8) 1 (a) Express in partial fractions. P(11 2P) A population of meerkats is being studied. The population is modelled by the differential equation dp 1 = P(11 2P), t 0, 0 < P < 5.5, dt 22 where P, in thousands, is the population of meerkats and t is the time measured in years since the study began. Given that there were 1000 meerkats in the population when the study began, (b) determine the time taken, in years, for this population of meerkats to double, (c) show that P = A B C where A, B and C are integers to be found. (Total for Question 16 is 12 marks) t 2 e 1, (6) 46

47 7. Differentiation 7. Differentiation Structure and calculation 7.1 Understand and use the derivative of f (x) as the gradient of the tangent to the graph of y = f (x) at a general point (x, y); the gradient of the tangent as a limit; interpretation as a rate of change sketching the gradient function for a given curve second derivatives differentiation from first principles for small positive integer powers of x and for sin x and cos x Understand and use the second derivative as the rate of change of gradient; connection to convex and concave sections of curves and points of inflection. 7.2 Differentiate x n, for rational values of n, and related constant multiples, sums and differences. 7.3 Apply differentiation to find gradients, tangents and normal maxima and minima and stationary points. Points on inflection. Identify where functions are increasing or decreasing. 7.4 Differentiate using the product rule, the quotient rule and the chain rule, including problems involving connected rates of change and inverse functions. 7.5 Differentiate simple functions and relations defined implicitly or parametrically, for first derivative only. 7.6 Construct simple differential equations in pure mathematics and in context, (contexts may include kinematics, population growth and modelling the relationship between price and demand). Sample questions AS Mathematics SAMs Paper 1 Q2 (7.1, 7.2, 7.3) The curve C has equation y = 2x Find the gradient of the curve at the point P(5, 6). (Solutions based entirely on graphical or numerical methods are not acceptable.) (Total for Question 2 is 4 marks) AS Mathematics SAMs Paper 1 Q6 (7.1) Prove, from first principles, that the derivative of 3x 2 is 6x. (Total for Question 6 is 4 marks) 47

48 7. Differentiation AS Mathematics SAMs Paper 1 Q16 (1.1, 7.1, 7.2, 7.3) A B 2x m E y m D C Figure 4 Figure 4 shows the plan view of the design for a swimming pool. The shape of this pool ABCDEA consists of a rectangular section ABDE joined to a semi-circular section BCD as shown in Figure 4. Given that AE = 2x metres, ED = y metres and the area of the pool is 250 m 2, (a) show that the perimeter, P metres, of the pool is given by P = 2x x 2x (4) 500 (b) Explain why 0 < x. (c) Find the minimum perimeter of the pool, giving your answer to 3 significant figures. (4) (Total for Question 16 is 10 marks) 48

49 7. Differentiation AS Mathematics SAMs Paper 1 Q15 (2.4, 6.3, 7.2, 7.3) y C1 Diagram not drawn to scale C2 O x The curve C1, shown in Figure 3, has equation y = 4x 2 6x The point P, 2 2 lies on C1. The curve C2, also shown in Figure 3, has equation y = 21 x ln (2x). The normal to C1 at the point P meets C2 at the point Q. Find the exact coordinates of Q. (Solutions based entirely on graphical or numerical methods are not acceptable.) (Total for Question 15 is 8 marks) A level Mathematics SAMs Paper 1 Q1 (1.1, 7.1, 7.2, 7.3) The curve C has equation y = 3x 4 8x 3 3. (a) Find (i) d y, dx 2 d y (ii) 2. dx (b) Verify that C has a stationary point when x 2 (b) Determine the nature of this stationary point, giving a reason for your answer. (Total for Question 1 is 7 marks) 49

50 7. Differentiation A level Mathematics SAMs Paper 1 Q10 (1.1, 5.6, 7.1) Given that is measured in radians, prove, from first principles, that the derivative of sin is cos You may assume the formula for sin (A B) and that, as h 0, sin h h 1 and cos h 1 0. h (Total for Question 10 is 5 marks) A level Mathematics SAMs Paper 1 Q13 (2.2, 3.1, 3.3, 5.3, 5.6, 7.2, 7.3, 7.5) The curve C has parametric equations x = 2 cos t, y = 3 cos 2t, 0 t π. (a) Find an expression for The point P lies on C where t dy in terms of t. dx 2. 3 The line l is the normal to C at P. (b) Show that an equation for l is 2x (2 3)y 1 = 0. The line l intersects the curve C again at the point Q. (5) (c) Find the exact coordinates of Q. You must show clearly how you obtained your answers. (6) (Total for Question 13 is 13 marks) 50

51 7. Differentiation A level Mathematics SAMs Paper 1 Q15 (2.9, 5.4, 7.2, 7.3, 7.4) Figure 5 Figure 5 shows a sketch of the curve with equation y f(x), where f(x) = 4sin 2x, 0 x e 2x 1 The curve has a maximum turning point at P and a minimum turning point at Q, as shown in Figure 5. (a) Show that the x-coordinates of point P and point Q are solutions of the equation tan 2x = 2. (4) (b) Using your answer to part (a), find the x-coordinate of the minimum turning point on the curve with equation (i) y f(2x), (ii) y 3 2f(x). (4) (Total for Question 15 is 8 marks) A level Mathematics SAMs Paper 2 Q3 (7.2, 7.4) Given y x(2x + 1) 4 dy, show that = (2x + 1) n (Ax + B), where n, A and B are constants to be dx found. (Total for Question 3 is 4 marks) 51

52 7. Differentiation A level Mathematics SAMs Paper 2 Q14 (1.1, 2.11, 7.2, 7.3) A company decides to manufacture a soft drinks can with a capacity of 500 ml. The company models the can in the shape of a right circular cylinder with radius r cm and height h cm. In the model they assume that the can is made from a metal of negligible thickness. (a) Prove that the total surface area, S cm 2, of the can is given by S = 2 r 2 + Given that r can vary, r (b) find the dimensions of a can that has minimum surface area. (5) (c) With reference to the shape of the can, suggest a reason why the company may choose not to manufacture a can with minimum surface area. (1) (Total for Questions 14 is 9 marks) A level Mathematics SAMs Paper 2 Q15 (3.1, 7.2, 7.3, 8.2, 8.3) Figure Figure 4 shows a sketch of the curve C with equation y = 5x 9x + 11, x 0. The point P with coordinates (4, 15) lies on C. The line l is the tangent to C at the point P. The region R, shown shaded in Figure 4, is bounded by the curve C, the line l and the y-axis. Show that the area of R is 24, making your method clear. (Solutions based entirely on graphical or numerical methods are not acceptable.) (Total for Question 15 is 10 marks) 52

53 8. Integration 8. Integration Structure and calculation 8.1 Know and use the Fundamental Theorem of Calculus 8.2 Integrate x n (excluding n = 1) and related sums, differences and constant multiples. Integrate e kx, x 1, sin kx,cos kx and related sums, differences and constant multiples. 8.3 Evaluate definite integrals; use a definite integral to find the area under a curve and the area between two curves 8.4 Understand and use integration as the limit of a sum. 8.5 Carry out simple cases of integration by substitution and integration by parts; understand these methods as the inverse processes of the chain and product rules respectively. (Integration by substitution includes finding a suitable substitution and is limited to cases where one substitution will lead to a function which can be integrated; integration by parts includes more than one application of the method but excludes reduction formulae.) 8.6 Integrate using partial fractions that are linear in the denominator. 8.7 Evaluate the analytical solution of simple first order differential equations with separable variables, including finding particular solutions (Separation of variables may require factorisation involving a common factor.) 8.8 Interpret the solution of a differential equation in the context of solving a problem, including identifying limitations of the solution; includes links to kinematics. Sample questions AS Mathematics SAMs Paper 1 Q3 (2.1, 2.2, 8.2, 8.3) Given that the point A has position vector 3i 7j and the point B has position vector 8i + 3j, (a) find the vector AB. (b) Find AB. Give your answer as a simplified surd. (Total for Question 3 is 4 marks) AS Mathematics SAMs Paper 1 Q5 (2.1, 2.2, 8.2, 8.3) 12 Given that f(x) = 2x x, x > 0, 2 2 show that 1 f ( x ) dx = (Total for Question 5 is 5 marks) 53

54 8. Integration A level Mathematics SAMs Paper 1 Q4 (6.4, 8.2, 8.3) Given that a is a positive constant and 2a a t 1 dt t = ln 7, show that a ln k, where k is a constant to be found. (Total for Question 4 is 4 marks) A level Mathematics SAMs Paper 1 Q14 (6.4, 8.3, 8.5, 9.3) Figure 4 Figure 4 shows a sketch of part of the curve C with equation y = x 2 ln 3 x 2x + 5, x > 0 The finite region S, shown shaded in Figure 4, is bounded by the curve C, the line with equation x = 1, the x-axis and the line with equation x = 3. The table below shows corresponding values of x and y, with the values of y given to 4 decimal places as appropriate. x y (a) Use the trapezium rule, with all the values of y in the table, to obtain an estimate for the area of S, giving your answer to 3 decimal places. (b) Explain how the trapezium rule could be used to obtain a more accurate estimate for the area of S. (1) 54

55 8. Integration (c) Show that the exact area of S can be written in the form b a + ln c, where a, b and c are integers to be found. (6) (In part (c), solutions based entirely on graphical or numerical methods are not acceptable.) (Total for Question 14 is 10 marks) A level Mathematics SAMs Paper 2 Q9 (2.3, 8.2, 8.3) Given that A is constant and 1 4 ( 3 x A) dx = 2A 2, show that there are exactly two possible values for A. (Total for Question 9 is 5 marks) A level Mathematics SAMs Paper 2 Q15 (3.1, 7.2, 7.3, 8.2, 8.3) Figure Figure 4 shows a sketch of the curve C with equation y = 5x 9x + 11, x 0. The point P with coordinates (4, 15) lies on C. The line l is the tangent to C at the point P. The region R, shown shaded in Figure 4, is bounded by the curve C, the line l and the y-axis. Show that the area of R is 24, making your method clear. (Solutions based entirely on graphical or numerical methods are not acceptable.) (Total for Question 15 is 10 marks) 55

56 8. Integration A level Mathematics SAMs Paper 2 Q16 (2.1, 6.3, 6.4, 8.6, 8.7, 8.8) 1 (a) Express in partial fractions. P(11 2P) A population of meerkats is being studied. The population is modelled by the differential equation dp 1 = P(11 2P), t 0, 0 < P < 5.5, dt 22 where P, in thousands, is the population of meerkats and t is the time measured in years since the study began. Given that there were 1000 meerkats in the population when the study began, (b) determine the time taken, in years, for this population of meerkats to double, (c) show that A P = B C t 2 e 1, (6) where A, B and C are integers to be found. (Total for Question 16 is 12 marks) 56